1. Measures of Location INFERENTIAL STATISTICS & DESCRIPTIVE STATISTICS Statistics of location Statistics of dispersion Summarise a central pointSummarises distribution around central point

2. Measures of Location ARITHMETIC MEAN Sum all observation, then divide by number of observations For a sample : For a population:

3. Measures of Location 0 2 4 6 8 10 12 14 16 18 3.54.55.56.57.58.59.510.511.5 Nightly Hours of Sleep No. of People X=7.07

4. Measures of Location MEDIAN 0 1 2 3 4 5 3334353637383940 Score Frequency For N = 15 the median is the eighth score = 37 Value that has equal no. of observations (n) on either side

5. Measures of Location MEDIAN 0 1 2 3 4 5 3334353637383940 Score Frequency For N = 16 the median is the average of the eighth and ninth scores = 37.5 Value that has equal no. of observations (n) on either side

6. Measures of Location MODE the most frequently occurring score value corresponds to the highest point on the frequency distribution For a given sample N=16: 33 35 36 37 38 38 38 39 39 39 39 40 40 41 41 45 The mode = 39 0 1 2 3 4 5 33343536373839404142434445 Score Frequency

7. Measures of Location AdvantagesDisadvantages Mode quick & easy to compute useful for nominal data poor sampling stability Median not affected by extreme scores somewhat poor sampling stability Mean sampling stability related to variance inappropriate for discrete data affected by skewed distributions Measures of central tendency Summary

8. Measures of Location DISPERSION These are measures of how the observations are distributed around the mean

9. Measures of Location DISPERSION: Range

10. Measures of Location DISPERSION: Variance ScoreDeviation Amy10-40 Theo20-30 Max30-20 Henry40-10 Leticia500 Charlotte6010 Pedro7020 Tricia8030 Lulu9040 SUM0 To see how ‘deviant’ the distribution is relative to another, we could sum these scores But this would leave us with a big fat zero mean = 50

11. Measures of Location DISPERSION: Variance So we use squared deviations from the mean, which are then summed This is the sum of squares (SS) ScoreDeviation Sq. of deviation Amy10-401600 Theo20-30900 Max30-20400 Henry40-10100 Leticia5000 Charlotte6010100 Pedro7020400 Tricia8030900 Lulu90401600 SUM06000 SS= ∑(X-X) 2

12. Measures of Location DISPERSION: Variance We take the “average” squared deviation from the mean and call it VARIANCE For a population: For a sample: (to correct for the fact that sample variance tends to underestimate pop variance)

13. Measures of Location DISPERSION: Standard deviation The standard deviation is the square root of the variance The standard deviation measures spread in the original units of measurement, while the variance does so in units squared. Variance is good for inferential stats. Standard deviation is nice for descriptive stats.

14. Measures of Location N = 28 X = 50 s 2 = 140.74 s = 11.86 N = 28 X = 50 s 2 = 555.55 s = 23.57 DISPERSION 0 2 4 6 8 10 12 14 0102030405060708090100 Scores # of People

15. Measures of Location DISPERSION Mean Variance Standard Deviation For a sample:For a population:

16. Measures of Location DISPERSION The Standard Error, or Standard Error of the Mean, is an estimate of the standard deviation of the sampling distribution of means, based on the data from one or more random samples e.g. 15 students each compile data sets of the heights of 20 people Numerically, it is equal to the square root of the quantity obtained when s squared is divided by the size of the sample. and n X = s s